I'm having trouble with this question on page 10 of V.I. Arnold's Math Methods book.
A mechanical system consists of three points. At the initial moment their velocities (in some inertial coordinate system) are equal to zero. Show that the points always remain in the plane which contained them at the initial moment.
This should be possible to prove using only the fact that Galilean transformations of space time send solutions of Newton's equation, $\ddot{\bf x}=f(\bf{x}, \dot{\bf{x}})$ to other solutions with different initial conditions.
I'd like to be able to say that $\ddot{\bf x}\cdot (u\times v)=0$ where $u$, $v$ are the relative positions from one particle, centered at the origin, to each of the others (assuming they aren't all three co-linear). Alternatively if I could show that for a rotation $G$ about the normal vector to the plane $(u\times v)$, $G(\ddot{\bf x}\cdot u)=\ddot{\bf x}\cdot u$ then clearly the acceleration is in the plane and the points cannot deviate from it. However I have not been able to make any headway with these attempts, I'd appreciate your suggestions.
Thanks